Summary: A DIXMIER-MOEGLIN EQUIVALENCE FOR
POISSON ALGEBRAS WITH TORUS ACTIONS
K. R. Goodearl
Abstract. A Poisson analog of the Dixmier-Moeglin equivalence is established for any affine
Poisson algebra R on which an algebraic torus H acts rationally, by Poisson automorphisms,
such that R has only finitely many prime Poisson H-stable ideals. In this setting, an additional
characterization of the Poisson primitive ideals of R is obtained they are precisely the prime
Poisson ideals maximal in their H-strata (where two prime Poisson ideals are in the same
H-stratum if the intersections of their H-orbits coincide). Further, the Zariski topology on
the space of Poisson primitive ideals of R agrees with the quotient topology induced by the
natural surjection from the maximal ideal space of R onto the Poisson primitive ideal space.
These theorems apply to many Poisson algebras arising from quantum groups.
The full structure of a Poisson algebra is not necessary for the results of this paper, which
are developed in the setting of a commutative algebra equipped with a set of derivations.
Motivated by existing and conjectured roles of Poisson structures in the theory of quan-
tum groups, we address some problems in the ideal theory of Poisson algebras. Recall,
for example, that Hodges and Levasseur [15, 16] and Joseph  have constructed bi-
jections between the primitive ideal space of the quantized coordinate ring Oq(G) of a
semisimple Lie group G and the set of symplectic leaves in G corresponding to a Poisson